The closest to exotic pet I can get is my rabbits.
Lucky you. The only thing that's been in my backyard was a moose several years ago.
Also, a lemur or ferret would be nice.
I can't really think of any besides the various reptiles I want to own when I retire or am rich and idle. I'd give them huge habitats. I want another iguana and another bearded dragon at some point, as well as a mali uromastyx, a Sudan plated lizard, a Savannah monitor lizard, and a gold tegu. Oh and a big turtle of some sort, but not too big unless it was potty trained (yes, you can potty train many types of reptiles).
I like seeing them happy in their habitats, though. If anything, I'd like to start an iguana sanctuary in my mom's native country of Honduras, where they're treated cruelly and eaten.